## There is only one mathematical object

• 447
A challenge to Platonism, which is IMO one of the more serious ones, is that mathematical objects lack clear identity conditions. What if there's only one object, Math, and the existence of individual objects is determined by whether they are validly derived from Math?

The logical problem here is determining what is a valid derivation. Naively, I say that a valid derivation is one that is logically coherent. So any logically consistent set of axioms from which one derives a mathematics is a valid derivation from Math, which means that those objects it posits are "real" in the sense that they are validly derived from math. However, they do not have independent existence, i.e. they are not separate entities in any sense but the conceptual. However, Math is mind-independent and our familiar mathematical objects are derivations therefrom.

If anybody in phil of math has already posited something like this, lemme know.
• 4.2k
I open my browser console and type Math period and get a list of mathematical functions and constants in that Math class. So at least for the Javascript predefined universe, this is the case. However, programmers add their own mathematical values and functions outside of Math all the time. Which could be analogous to how we invent new branches of math.

But maybe those maths already exist in the possible space of all Math defined by inherent logic. We could say all possible games of chess or life worlds similarly exist. That requires a realist commitment to possible worlds or at least possibility space. But what happens when one modifies the logic? Does that cause a new possibility space to come into existence? You can always create derivatives of chess, or your own cellular automata.

Or the case of math, add new concepts like infinity or imaginary numbers with their own derivation of the rules.
• 79
I know nothing about philosophy of math but I'll take a shot at this.

All you establish with
logically consistent set of axioms from which one derives a mathematics is a valid derivation from Math

is that it is possible for this object to be real. It exists in the sense that you can imagine it as a unity (a single object).

I'll set aside the obvious objection that each of these derived-from-math objects would imply there's more than one object, since you seem to think they are really just aspects of the single object, Math, as it appears in different ways.

By the way, some countries in the world don't say "Math" they say "Maths".
• 447

If it helps: I'm primarily interested in answering the identity-condition objection to Platonism. The dialectic goes kinda like this, where P is a Platonist and AP is an anti-Platonist:

P: The triangle is a mathematical object that exists.
AP: All of the triangles, or just one?
P: All of them.
AP: Okay. So how do we tell the difference between two triangles? Do all acute triangles answer to the form of the acute triangle, or are there multiple acute triangles?

And so on. The two big objections to Platonism that arise from conversations like this are that Platonic objects lack clear identity conditions and that the ontology is profligate, a crowded slum, what Quine called Plato's Beard. Reducing every object to Math should answer both objections.

But what happens when one modifies the logic?

This is a good question. I would say that modifications to the logic introduce new subdomains. Provided that translation functions can be constructed between those domains, there shouldn't be any problem with all of this being and expression of Math.

EDIT: or perhaps, instead of translation functions, we can say that any logical space that gives us valid derivations from Math is constructed via identity statements.
• 8.3k
I'm in unfamiliar territory but does the type-token distinction seem relevant to the OP? Speaking in terms of the geometrical object triangle, there's a type triangle and all other triangles are its tokens. Quine's notion of Plato's beard seems to ignore/overlook this.
• 8.1k
I'm in unfamiliar territory but does the type-token distinction seem relevant to the OP? Speaking in terms of the geometrical object triangle, there's a type triangle and all other triangles are its tokens. Quine's notion of Plato's beard seems to ignore/overlook this.

I think the point is that there are numerous different types of triangles. And if you want to argue that there is only one type, "the triangle", then why isn't the triangle just one type of polygon? And the polygon is a type of geometrical figure, and so on.
• 10.9k
A challenge to Platonism, which is IMO one of the more serious ones, is that mathematical objects lack clear identity conditions.

I would have thought that the identity conditions of integers was abundantly obvious. I mean, any integer is distinct from all other integers - how does that not constitute an 'identity condition'?

As for the triangle, it's 'a flat plane bounded by three intersecting straight lines'. That applies to any triangle. The 'form' is not the shape.
• 979
You are thinking too small. One mythical triangle, a hundred triangles, immaterial. Consider a structure akin the Tegmark's mathematical universe that interweaves and supports the entire universe. Call this MATH.

We mathematicians pull threads from the rich fabric, imagining ourselves creating mathematics, when in fact we only uncover wisps of a majestic and largely unknowable tapestry - a single entity beyond our wildest speculations.
• 79
AFIAK the typical translational convention is to use "same" for situations where Plato compares two things, while "identity" refers to a single object. But I think Plato is mistaken in distinguishing these two notions from each other. There is no such thing as identity as Plato conceives of it because it has no function.
• 3.8k
Tegmark's mathematical universe

:100:

The concrete world is an abstract object: it's just the one that we're a part of.

Because you can make abstract objects from collections of other abstract objects, then yes OP, you can say that there is just one abstract object of which all other objects are parts, and that one all-encompassing abstract object is the entirety of existence in the broadest possible sense.
• 8.1k
I would have thought that the identity conditions of integers was abundantly obvious. I mean, any integer is distinct from all other integers - how does that not constitute an 'identity condition'?

How can we say that 2 represents a unique and particular object? That is the identity condition. To have an identity is to be identifiable as a unique and particular individual. But every time that there are two objects, the number 2 is represented, so 2 represents something universal, rather than something unique and particular. Therefore it appears like the number signified by the numeral 2 cannot fulfill identity conditions.

As for the triangle, it's 'a flat plane bounded by three intersecting straight lines'. That applies to any triangle. The 'form' is not the shape.

We can attempt to provide identity through the means of a definition, but the definition always allows that more than one thing can be identified as fulfilling the identity conditions of the definition. So a definition cannot serve to give us adequate identity conditions because it allows that more than one thing might have the same identity.
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The concrete world is an abstract object: it's just the one that we're a part of.

Because you can make abstract objects from collections of other abstract objects, then yes OP, you can say that there is just one abstract object of which all other objects are parts, and that one all-encompassing abstract object is the entirety of existence in the broadest possible sense.
:100:

Sub specie aeternitatis: natura naturans. (Spinoza)
• 10.9k
How can we say that 2 represents a unique and particular object? That is the identity condition.

Right. So 'identity condition' pertains to individual identity, something unique and particular. What is the source or definition of 'identity condition'?

We can attempt to provide identity through the means of a definition, but the definition always allows that more than one thing can be identified as fulfilling the identity conditions of the definition.

Be that as it may, a triangle will never have other than three sides.

BTW there is a current article in Smithsonian Magazine about the ongoing appeal of mathematical platonism. https://www.smithsonianmag.com/science-nature/what-math-180975882/ . It says while there is some support for mathematical platonism, that:

Other scholars—especially those working in other branches of science—view Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

which basically says it all.
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Platonic objects lack clear identity conditions

I don't know what that means. Your example was triangles. Triangles are identified up to similarity by their angles; and up to congruence by the lengths of their sides; and identified uniquely by their congruence class and position and orientation in space. It's perfectly simple to define a similarity class of triangles or a congruence class of triangles or a particular triangle. I don't follow your examples.

A classic Platonic object in math is the unit circle. It's a circle of radius one centered at the origin. Now you're right, coordinate systems are arbitrary so in fact we could locate the unit circle anywhere on the plane. In that sense "the" unit circle is arbitrary. But once you fix a rectangular coordinate system, you have a unique unit circle and you can then define the trigonometric functions, Fourier series, topological groups, and all the rest of the interesting mathematical concepts that generalize or abstract from the unit circle.

Math isn't concerned at all with particular objects; only with abstract forms. It doesn't matter what numbers "are," only how they behave and how they relate to other numbers. This is mathematical structuralism. As David Hilbert said, "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs."

That quote answers your concern that we can't identify particular things. We don't care about identifying particular things. Rather, we only care about structural and logical relationships among particular things; and those relations are independent of the things themselves.

To those throwing rocks at mathematical Platonism (as I do when I'm taking the other side of this debate), was 3 prime before there were intelligent life forms in the universe? If that's too easy (I don't think it is), were there infinitely many primes? Or at least no largest prime?
• 8.1k
Right. So 'identity condition' pertains to individual identity, something unique and particular. What is the source or definition of 'identity condition'?

I don't know where Pneumenon gets this specific terminology, but I know the identity condition as the "law of identity", which states that a thing is the same as itself. Leibniz' posited the "identity of indiscernibles" which stipulates that each individual thing is unique.

Be that as it may, a triangle will never have other than three sides.

That's true, but "triangle" does not suffice as a thing's identity because many things are said to be triangles. What I believe to be Pneumenon's point, is that whatever "triangle" refers to, it cannot be a thing, because all things have a distinct and unique identity according to the law of identity, or what Pneumenon calls "identity condition". Pneumenon uses this argument against any Platonists who believe that ideas exists as objects, arguing that ideas such as 'triangle" do not fulfil the "identity conditions to be rightly called "objects".

This is an ontological issue, and I don't think it's meant as a simple attack against dualism. When using this argument we do not mean to imply that immaterial things do not qualify as "objects", therefore we ought to reject the existence of the immaterial. What I believe, is that the argument is meant to elucidate the fundamental, and radical difference between the immaterial and the material, demonstrating that it is a basic misunderstanding to portray the immaterial as objects.

If we accept this perspective, that the immaterial ought not be represented as consisting of objects, there is far reaching consequences. Many mathematical axioms such as those of set theory rely on the assumption of mathematical objects. Further, we can see the trend in physics, to translate the immaterial wave fields to objects (particles), though things like virtual particles appear as immaterial. I see this reduction of the immaterial to objects, as a real problem. It basically says that the immaterial has to be like the material, existing as objects. But when we see that the immaterial does not fulfill the "identity conditions" which is required for existence as objects, we need to move on and recognize that the immaterial is radically different from the assumption of immaterial objects.
• 10.9k
I know the identity condition as the "law of identity", which states that a thing is the same as itself.

But the most succinct formulation of 'the law of identity' is 'a=a'. So are you saying that 'a' doesn't have an identity?

"triangle" does not suffice as a thing's identity because many things are said to be triangles.

Many things can be triangles, but that is only insofar as those things assume that form. The form itself is not a thing. Only three-sided flat planes bounded by lines constitute a triangle but that covers an endless variety of things. That's the 'thing' about universals. ;-)

--

To those throwing rocks at mathematical Platonism (as I do when I'm taking the other side of this debate), was 3 prime before there were intelligent life forms in the universe?

That's a very interesting question. I generally defend mathematical Platonism, on the grounds that numbers are real but immaterial, in that they can only be grasped by a rational mind, but they're the same for anyone who can count.

But if this is true, then it falsifies the idea that only material things are real; hence the passage I quoted above from the Smithsonian article. Platonism is generally resisted on exactly those kinds of grounds. Hence fictionalism etc.

There was an essay by Paul Benacereff, Mathematical Truth, which has apparently been quite influential, pointing out that mathematical intuition is not empirically verifiable, and so philosophically at odds with empiricism. In an analysis from an article in IEP we read:

[Rationalist philosophers] claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies. ...The indispensability argument in the philosophy of mathematics is an attempt to justify our mathematical beliefs about abstract objects, while avoiding any appeal to rational insight. Its most significant proponent was Willard van Orman Quine.

I find it amusing that modern philoosphy has to come up with arguments (or tie itself in knots) to find an alternative to rational insight, especially as so many popular intellectuals blather endlessly about the 'rationalism of science'.

In any case, getting back to your question: my solution to it is that in humans, the mind has evolved to the point where it can grasp the nature of prime numbers. But that means neither that prime numbers are 'out there somewhere' (which is naturalism, again) or 'in the mind' (which is subjectivism and relativism). They can only be grasped by a mind, but they're the same for all minds. Which is pretty much a definition of 'objective idealism'. We discover the deep laws and regularities which 'govern' how phenomena behave, but science itself can't explain those laws - as you said elsewhere, science 'describes but doesn't explain' on that level. If that was understood it would solve many a philosophical conundrum.

That is why I think mathematical Platonism still has a pretty strong hand.
• 8.1k
But the most succinct formulation of 'the law of identity' is 'a=a'. So are you saying that 'a' doesn't have an identity?

You know that this is just a symbolic representation of the law. So the symbols need to be interpreted. What 'a=a' represents is that for any object, represented as 'a', that object is the same as itself.

Therefore it's not saying that 'a' doesn't have an identity, it's saying that the identity of the object represented by 'a', is the object represented by 'a'. In other words, an object is its own identity. Aristotle found it necessary to formulate the law of identity in this way, to recognize the difference between the identity we assign to an object, and the object's true identity. Sophistry had demonstrated that the identity which we give an object is sometimes incorrect. So we need a way to allow that the human assigned identity is incorrect, yet the object still has a true identity which the human beings have not determined. Therefore Aristotle posited that the identity of any object is within itself.

Many things can be triangles, but that is only insofar as those things assume that form. The form itself is not a thing. Only three-sided flat planes bounded by lines constitute a triangle but that covers an endless variety of things. That's the 'thing' about universals.

Yes, that's a feature of universals. the word "triangle" covers an endless variety of things. But this same feature indicates to us, that a universal is not itself an individual thing. A universal is not a thing because it does not conform to the law of identity. It cannot be identified as a thing, because it has no identity as a thing. But, a thing necessarily has an identity, itself. Therefore a universal is not a thing.

This does not imply that there is no immaterial existence, it just means that in our understanding of immaterial existence we have to get beyond the idea that immaterial existence is in the form of things. That is just a sort of mistake which has developed from human beings relating what they know of the material world, to the immaterial, in an attempt to understand the immaterial. Because the material world consists of things, we want to apply the same principles to the immaterial, and portray the immaterial world as consisting of things. But the law of identity is interjected to demonstrate to us, the fundamental difference between material and immaterial, and that this would be a misunderstanding.

To those throwing rocks at mathematical Platonism (as I do when I'm taking the other side of this debate), was 3 prime before there were intelligent life forms in the universe? If that's too easy (I don't think it is), were there infinitely many primes? Or at least no largest prime?

The word "prime" was created by human beings, and has a meaning according to what human beings think. It does not make any sense at all to ask about the meaning of the word "prime", or any word for that matter, at a time before the word existed. (What did the word "prime" mean before it existed?) I'm sure you can understand that. We can however use the word to refer to something that we believe existed before the word. Like "earth" for example is supposed to have existed before the word. Your question therefore asks, whether there was something which we refer to with "3", and something which we refer to with "prime", which existed prior to the existence of these words, and that is a difficult metaphysical question without a straight forward answer.
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It does not make any sense at all to ask about the meaning of the word "prime", or any word for that matter, at a time before the word existed.

So, you're a relativist after all?
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Many mathematical axioms such as those of set theory rely on the assumption of mathematical objects

Can you name one such? Structuralism is in these days. It doesn't matter if you call sets "beer mugs" as Hilbert pointed out. It's the properties and relations that matter, not the nature of individual things.

whether there was something which we refer to with "3", and something which we refer to with "prime", which existed prior to the existence of these words, and that is a difficult metaphysical question without a straight forward answer.

Hey, something on which we agree!
• 8.1k
So, you're a relativist after all?

No, not necessarily, I just recognize that it is impossible for a word to have a meaning before that word exists.

Can you name one such?

I think we've been through this before. You insisted on an unreasonable separation between "objects" and "mathematical objects", such that mathematical objects are not a type of object.

We could start with the axiom of extensionality. Any axiom which treats numbers as elements of a set, treats the numbers as objects.

It doesn't matter if you call sets "beer mugs" as Hilbert pointed out. It's the properties and relations that matter, not the nature of individual things.

The issue though, is that set theory treats them as "individual things", therefore Platonism is implied. Set theory relies on Platonism because it cannot proceed unless what 2, 3, 4, refer to are objects, which can be members of a set.
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I think we've been through this before.

Good point, I knew better than to start. Don't know what I'm thinking. This can't end well.

You insisted on an unreasonable separation between "objects" and "mathematical objects", such that mathematical objects are not a type of object.

Well, mathematical objects are abstract objects. But I agree that numbers aren't like rocks. That doesn't mean that numbers don't exist. It only means that numbers are abstract. And, per structuralism and Benacerraf's famous essay, What Numbers Cannot Be, numbers are not any particular thing. They're not actually sets, even though they are typically represented as sets. Numbers are the abstract things represented by sets. I suspect you and I might be in violent agreement on this point, but I'm not sure.

We could start with the axiom of extensionality. Any axiom which treats numbers as elements of a set, treats the numbers as objects.

Only sets can be elements of sets in pure set theory. So we can represent numbers by sets and say that the $\in$ relation holds between a pair of sets; but you are reading more into that than is intended. In high school set theory we would say that the students are members of the school, and we'd call the school a set and the students elements of that set. And I doubt that you'd disagree. But i formal set theory we are not reifying, I think that's the word, the things that are elements of sets. We're just saying that the membership relation holds between the abstract things represented by the symbols. You can, if you like, view the entire enterprise as an exercise in formal symbol manipulation that could be carried out by computer, entirely devoid of meaning. It would not make any difference to the math.

The issue though, is that set theory treats them as "individual things", therefore Platonism is implied. Set theory relies on Platonism because it cannot proceed unless what 2, 3, 4, refer to are objects, which can be members of a set.

Maybe you'd like category theory better. In categorical set theory there are no elements at all, only relationships among sets. But in category theory they call things "objects" and that might make you unhappy.

We could go down a rabbit hole here but just tell me this. Do you believe that if E is the set of even positive integers, then E = {2, 3, 4, 6, ...}. Do you agree with that statement? Or do you deny the entire enterprise? I'm trying to put a metric on your mathematical nihilism. Is it naughty to put numbers into sets? Why? Some numbers are prime, some are even, some are solutions to various equations. Why can't we collect them into sets? Either conceptually or, if you don't like that, formally?
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I just recognize that it is impossible for a word to have a meaning before that word exists.

Your question therefore asks, whether there was something which we refer to with "3", and something which we refer to with "prime", which existed prior to the existence of these words, and that is a difficult metaphysical question without a straight forward answer.

The Platonist explanation is that these 'things' - they're not actually things, which is part of the point - are discerned by the rational intellect, nous. They transcend individual minds, but they're constituents of rational thought because thought must conform to them in order to proceed truly. In which case they precede their discovery by the mind, not in a temporal sense, but in the sense that they must already be the case in order for thought to be rational in the first place.

Why this is so murky, and so controversial, is because it all harks back to the disputes between realists and nominalists in the late medieval period. Nominalism was the chief precursor to today’s empiricism, and it has so permeated the public discourse that we don’t know how to think any other way. History is written by the victors, and they were indubitably victorious in this matter. That’s why discussion of universals generally only draws blank stares in modern culture. And also because it thinks ‘mind’ is the product of undirected matter and that number is ‘a product’ of this undirected process.
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Well, mathematical objects are abstract objects. But I agree that numbers aren't like rocks. That doesn't mean that numbers don't exist. It only means that numbers are abstract. And, per structuralism and Benacerraf's famous essay, What Numbers Cannot Be, numbers are not any particular thing. They're not actually sets, even though they are typically represented as sets. Numbers are the abstract things represented by sets. I suspect you and I might be in violent agreement on this point, but I'm not sure.

Well, I wouldn't go so far as to call it a violent agreement. The point of the op I believe, is that it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity. And so, if we start talking about them as if they are objects, and believe that they have identities as objects, and treat them that way, when they do not, there is bound to be problems which arise.

Where we agree is that they are "abstract", but the problem is in where we go from here.

We're just saying that the membership relation holds between the abstract things represented by the symbols.

Here is where the difference between us appears to arise. You are saying that there are "abstract things represented by the symbols". That's Platonism plain and simple, the "abstract things" are nothing other than Platonic Ideas, or Forms. See, you even allow that there are relations between these things. But from my perspective, a symbol has meaning, and meaning is itself a relation between a mind and the symbol. So I see that you've jumped to the conclusion that this relation between a symbol and a mind, is itself a thing, and you then proceed to talk about relations between these supposed things which are really just relations, and not things at all, in the first place.

You can, if you like, view the entire enterprise as an exercise in formal symbol manipulation that could be carried out by computer, entirely devoid of meaning. It would not make any difference to the math.

If you can follow what I said above, then I'll explain why there's a real problem here. The relation between a symbol and a mind, which is how I characterized the abstract above, as meaning, is context dependent. When you characterize this relation, the abstract, as a thing, you characterize it as static, unchangeable. This is what allows you to say that it is the same as manipulating symbols devoid of meaning, the symbol must always represent the exact same thing. But here is where this thing represented, the abstract, fails the law of identity, the meaning, which is the relation between the mind and the symbol, is context dependent and does not always remain the exact same.

We could go down a rabbit hole here but just tell me this. Do you believe that if E is the set of even positive integers, then E = {2, 3, 4, 6, ...}. Do you agree with that statement? Or do you deny the entire enterprise? I'm trying to put a metric on your mathematical nihilism.

I think we've discussed this enough already, for you to know that I denounce all set theory as ontologically unsound, fundamentally. It doesn't mean a whole lot though, only that I think it's bad, like if I saw a bunch of greedy people behaving in a way I thought was morally bad, I might try to convince them that what they were doing is bad. However, if it served them well, and made their lives easy, I'd have a hard time convincing them.

The Platonist explanation is that these 'things' - they're not actually things, which is part of the point - are discerned by the rational intellect, nous.

I think that modern Platonism treats the abstract as things. This is what allows them to define static relations between these things, as fishfry describes in set theory. But if the abstract is not really things, then this static nature is unsupported, and so are the static relations unsupported.

They transcend individual minds, but they're constituents of rational thought because thought must conform to them in order to proceed truly.

It appears, that what you are saying here is that there are some sort of ways of thinking, which thought must conform to in order to be rational. I would agree, but the specifics of the correct way is something to be determined. So I will ask you a question to see if you might have an answer. Wouldn't the way of thinking, which would be judged as the rational way, be itself dependent on and therefore determined by the particular situation, or context? So for example, there is some sort of way of thinking which is the correct way for a particular situation, and this transcends all individual minds, making it the correct way for any mind, in that situation. But that correctness, and the way of thinking itself which is correct, is determined by and specific to the particular situation itself. Would you agree with this? And what about the inverse? Is it possible that there is one correct way of thinking which transcends all situations as the correct way of thinking no matter what the situation?
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Wouldn't the way of thinking, which would be judged as the rational way, be itself dependent on and therefore determined by the particular situation, or context? So for example, there is some sort of way of thinking which is the correct way for a particular situation, and this transcends all individual minds, making it the correct way for any mind, in that situation. But that correctness, and the way of thinking itself which is correct, is determined by and specific to the particular situation itself. Would you agree with this?

But, the point about universals is that they're universally applicable, isn't it? They're applicable in any context. Think about scientific laws, which I think must in some sense be descended from such ideas. Water doesn't sometimes flow uphill, for instance. Think also about Kant's deontological ethics, which individuals are obliged to conform to if their actions are to be ethically sound.

Furthermore, I can't arbitarily designate the rules of math or the laws of logic, I have to conform to them, as much as I'm able (which in my case, is not very much). I can adapt them to my situation, I can use them to advantage, but I can't change them. (Again, the clearest exposition of these ideas are in the Cambridge Companion to Augustine, on the passage on Intelligible Objects.)

I think that modern Platonism treats the abstract as things.

Modern thought treats everything as a thing. (Who's paper is it, 'What is a thing'? Heidegger, I think.) Anyway, the point is, the modern mentality is so immersed in the sensory domain, that it can only reckon in terms of 'things'. Things are 'what exists' - which is what throws us off about mathematical concepts, they're not things, but they seem real, so 'what kind' of reality do they have? In our world, real things can only be 'out there', the only alternative being 'in the mind'. But in reality, 'out there' and 'the mind' are not ultimately separable - hence, as I say, the logic of objective idealism. But it takes a shift in perspective to see it.
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Well, I wouldn't go so far as to call it a violent agreement. The point of the op I believe, is that it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity.

I confess I didn't understand the OP at all. They seemed to be saying that there is only one mathematical object, all of math. I couldn't parse that. I didn't join the thread till you were talking about something I could at least understand.

But now you say, "it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity." When a while back you disagreed that 2 + 2 and 4 represent the same mathematical object (regarding which you are totally wrong but nevermind), that was one thing. But now you seem to be saying that 4 = 4 is not valid to you because mathematical objects don't fulfill the law of identity. Am I understanding you correctly? Do you agree that 4 = 4 and that both sides represent the same mathematical object? Or are you saying that since there aren't any mathematical objects, 4 = 4 does not represent anything at all?

And so, if we start talking about them as if they are objects, and believe that they have identities as objects, and treat them that way, when they do not, there is bound to be problems which arise.

4 = 4 is true by the law of identity, yes or no? I can't believe I'm even having this conversation. You've never convinced me that your mathematical nihilism isn't an elaborate troll.

Where we agree is that they are "abstract", but the problem is in where we go from here.

Probably nowhere, as I knew this would. I imagine you knew it too.

Here is where the difference between us appears to arise. You are saying that there are "abstract things represented by the symbols". That's Platonism plain and simple, the "abstract things" are nothing other than Platonic Ideas, or Forms. See, you even allow that there are relations between these things.

On my Platonist days I say that. But if you object, I am perfectly willing to do exactly the same math, but regarding it as a purely formal game of symbol manipulation, no different in principle than chess. Then you have no philosophical objections, any more than you would to formal games like chess or Go or Parcheesi. 4 + 4 and 2 + 2 = 4 are legal moves in my game. It doesn't matter to me. Do you at least accept that math can be regarded as a formal game without regard to meaning? It's actually often helpful to think of it that way even if you secretly believe otherwise. One can be pragmatic regarding one's philosophy.

But from my perspective, a symbol has meaning,

Well then you are the Platonist. What is the meaning of the way the knight moves in chess? Clearly there is no real world referent. Nor is there any real world referent for many of the constructs of higher math. I'm willing to stipulate, for purposes of this discussion, that there are no real world referents for any of the constructs of math. So what? As long as the rules are consistent and the game is fun, we can all play.

and meaning is itself a relation between a mind and the symbol. So I see that you've jumped to the conclusion that this relation between a symbol and a mind, is itself a thing,

No, you are the one saying that. I'm saying that if you don't believe 4 represents an abstract mathematical object, then it's perfectly ok to regard it as a meaningless symbol subject to the laws of arithmetic, which can be mindlessly encoded in a computer program like your calculator. When you punch '4' into your pocket calculator, the circuitry doesn't know what 4 means, but it perfectly manipulates 4 according to the rules with which it's programmed.

and you then proceed to talk about relations between these supposed things which are really just relations, and not things at all, in the first place.

Ok fine, it's all a meaningless formal game. It makes no difference to me. But if you don't think 4 is a thing, you are most definitely a mathematical nihilist. When you go to the store and buy a dozen eggs, do you make these same points at the checkout stand?

If you can follow what I said above, then I'll explain why there's a real problem here.

Truly I stopped following you back when you claimed that 2 + 2 and 4 don't represent the same mathematical object, when in fact they do. And when I gave you a purely formal syntactic proof that they represent the same thing, and you refused to even engage with my argument. You didn't say, "I reject the Peano axioms," or "I have it on good authority that Giuseppe Peano cheated at cribbage and is therefore not to be trusted," or "You made a mistake on line 3," or anything like that. You simply ignored the argument entirely, despite my asking you several times to respond. You have yet to demonstrate that you're having a serious conversation with me.

The relation between a symbol and a mind, which is how I characterized the abstract above, as meaning, is context dependent. When you characterize this relation, the abstract, as a thing, you characterize it as static, unchangeable.

I have no trouble with time-dependent assignment of meaning. But, are you claiming that 4 means one thing to you today and other thing tomorrow? What ever are you talking about?

This is what allows you to say that it is the same as manipulating symbols devoid of meaning, the symbol must always represent the exact same thing.

Well yes, that's your nihilism speaking again. If I say 4 = 4 and you assert that the symbol '4' may have different meanings on each side of the equation, you are the crazy one. I don't mean that in pejorative sense, it's an accurate description.

Of course a symbol like 'x' may mean one thing in one context and a different thing in another, but I truly hope you are not thinking that this is a very deep or significant point. Within a given context, a symbol only has one particular meaning; otherwise we can't do math, we can't do science, we can't even get on the bus. "Oh, today the #4 bus goes to Liverpool. You must want yesterday's #4 bus that went to Bristol." Come on, how can you expect me to take you seriously when you assert such nonsense?

But here is where this thing represented, the abstract, fails the law of identity, the meaning, which is the relation between the mind and the symbol, is context dependent and does not always remain the exact same.

In the string 4 = 4, does the symbol '4' refer to the same thing on each side of the equation? Is this or is this not an instance of the law of identity?

I think we've discussed this enough already, for you to know that I denounce all set theory as ontologically unsound, fundamentally.

Well ok you're a mathematical nihilist and you don't deny it. But I doubt you actually live like that. You couldn't pay your bills or read the paper, if the meaning of the symbols keeps changing for you.

It doesn't mean a whole lot though, only that I think it's bad, like if I saw a bunch of greedy people behaving in a way I thought was morally bad, I might try to convince them that what they were doing is bad.

Set theorists are morally bad people? Who need to be shown the error of their ways? Wow you are far gone my friend. Are you a type theorist? A category theorist? Or do you feel that those people are morally bad too? Not just wrong, but morally bad. I'm genuinely curious about this. Is it just set theory? Or do you feel this way about all historical attempts at mathematical systemization and formalization, from Euclid on down to the present?

However, if it served them well, and made their lives easy, I'd have a hard time convincing them.

You haven't convinced me that you're serious about anything you write. At least when you converse with me.

Ok to make this short, can you please just respond to these two questions:

* Is 4 = 4 an instance of the law of identity; or does the symbol '4' have a different meaning on each side?

* Is Euclid morally bad by virtue of attempting mathematical synthesis?
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But, the point about universals is that they're universally applicable, isn't it? They're applicable in any context. Think about scientific laws, which I think must in some sense be descended from such ideas. Water doesn't sometimes flow uphill, for instance. Think also about Kant's deontological ethics, which individuals are obliged to conform to if their actions are to be ethically sound.

I agree, but the universality of universals is exactly what makes them incompatible with identity which is what particulars have. This produces a hole, or gap in human understanding, because the material reality, to which we apply these universals in our attempts to understand, consists of unique particulars. This means that there is always a deficiency in human understanding. The point in enforcing a "law of identity", is to recognize and adhere to this understanding, that this gap exists, so that we do not push Platonism to its extremes, claiming that the physical universe is composed of mathematical objects. This is impossible, because mathematical objects are universals, but the universe is composed of unique particulars. The gap of incompatibility between these two demonstrates that such extreme Platonism, better known as Pythagorean Idealism, cannot be true.

Furthermore, I can't arbitarily designate the rules of math or the laws of logic, I have to conform to them, as much as I'm able (which in my case, is not very much). I can adapt them to my situation, I can use them to advantage, but I can't change them. (Again, the clearest exposition of these ideas are in the Cambridge Companion to Augustine, on the passage on Intelligible Objects.)

This goes both ways. There are people who literally make up, or create axioms of mathematics, it's what fishfry calls pure mathematics. We must ensure that the mathematical axioms which we employ conform to reality or else they will lead us astray. Therefore it is actually necessary that we do change mathematical axioms as we try and test them. And if you look at the history of them you will see that they evolve, just like knowledge evolves, and living beings evolve. This implies that we must accept such things as evolving properties of living beings, rather than eternal immutable objects.

That the rules of math, and laws of logic appear to you as something which you cannot change, is the result of many years of usage by many different people. They are tried and tested so they are what we use. If, in your occupation there are rules which must be applied in order for you to fulfill your job, then you cannot change those rules or else your job would not get done. However, the world is full of innovative and creative people who could come up with new rules, which fulfill an end different from what you are doing, but is judged to be better than yours, and renders yours obsolete. Therefore this idea of "I can't change them" is just an illusion. Yes, if you want to keep doing what you are doing, you cannot change them, but if you quit what you are doing, and adopt other rules which are conducive to something else instead, which renders what you were doing before as obsolete, you really do change them.

Modern thought treats everything as a thing. (Who's paper is it, 'What is a thing'? Heidegger, I think.) Anyway, the point is, the modern mentality is so immersed in the sensory domain, that it can only reckon in terms of 'things'. Things are 'what exists' - which is what throws us off about mathematical concepts, they're not things, but they seem real, so 'what kind' of reality do they have? In our world, real things can only be 'out there', the only alternative being 'in the mind'. But in reality, 'out there' and 'the mind' are not ultimately separable - hence, as I say, the logic of objective idealism. But it takes a shift in perspective to see it.

This I completely agree with, but again we have to be aware of where things go the other way. Philosophers looking toward the reality of Ideas describe them in terms of objects to facilitate understanding. However, we can see that thinking is an activity and it doesn't really exist as objects. On the other side of the coin, we see physicists who look at objects and use mathematical ideas to describe them in terms of activity. So we can see that the world of physical objects gets reduced to the activity of energy, because this is compatible with the realm of thought, mathematics. Now we have no adequate principles to separate objects from activities, and it's an ontological mess.

But now you say, "it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity." When a while back you disagreed that 2 + 2 and 4 represent the same mathematical object (regarding which you are totally wrong but nevermind), that was one thing. But now you seem to be saying that 4 = 4 is not valid to you because mathematical objects don't fulfill the law of identity. Am I understanding you correctly? Do you agree that 4 = 4 and that both sides represent the same mathematical object? Or are you saying that since there aren't any mathematical objects, 4 = 4 does not represent anything at all?

I guess you don't remember the key points (from my perspective) of or previous discussions. What I objected to was calling things like what is represented by 4, as "objects". I made this objection based on the law of identity, similar to the op here. You insisted it's not an "object" in that sense of the word, it's a "mathematical object". And I insisted that it ought not be called an object of any sort. So you proceeded with an unacceptable interpretation of the law of identity in an attempt to validate your claim. What I believe, is that "mathematical object" is an incoherent concept.

4 = 4 is true by the law of identity, yes or no?

This depends on what = represents. Does it represent "is the same as", or does it represent "is equal to"? From our last discussion, you did not seem to respect a difference between the meaning of these two phrases. If you're still of the same mind, then there is no point in proceeding until we work out this little problem. This is why I say context of the symbol is important. When the law of identity is represented as a=a, = symbolizes "is the same as". But when we write 2+2=4, = symbolizes "is equal to". If we assume that a symbol always represents the very same thing in every instance of usage, we are sure to equivocate. Clearly, "is the same as" does not mean the same thing as "is equal to".

Do you at least accept that math can be regarded as a formal game without regard to meaning?

No, of course not, that's clearly a false representation of what math is. That would be like saying that 2+2=4 could be considered to be valid regardless of what the symbols mean. That's nonsense, it's what the symbols mean which gives validity to math.

But, are you claiming that 4 means one thing to you today and other thing tomorrow?

Yes, that is exactly the case it can even mean something different in the same sentence. When some says "I reserved a table for 4 at 4", each instance of 4 means something different to me. And, as I explained to you already, when someone says 2+2=4, each instance of 2 must refer to something different or else there would not be four, only two distinct instances of the very same two, and this would not make four.
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The point in enforcing a "law of identity", is to recognize and adhere to this understanding, that this gap exists, so that we do not push Platonism to its extremes, claiming that the physical universe is composed of mathematical objects. This is impossible, because mathematical objects are universals, but the universe is composed of unique particulars.

This is the point of hylomorphic dualism, is it not?

if the proper knowledge of the senses is of accidents, through forms that are individualized, the proper knowledge of intellect is of essences, through forms that are universalized. Intellectual knowledge is analogous to sense knowledge inasmuch as it demands the reception of the form of the thing which is known. But it differs from sense knowledge so far forth as it consists in the apprehension of things, not in their individuality, but in their universality.

Thomistic Psychology: A Philosophic Analysis of the Nature of Man, Brennan.

Clearly, "is the same as" does not mean the same thing as "is equal to".

I say it does. I think you're splitting hairs for the sake of argument.
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Set theorists are morally bad people? Who need to be shown the error of their ways?

Good one! :rofl:

This might be the key to their salvation: { } = {N} :cool:
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I guess you don't remember the key points (from my perspective) of or previous discussions.

My mind has blocked them out as traumatic experiences.

What I objected to was calling things like what is represented by 4, as "objects". I made this objection based on the law of identity, similar to the op here. You insisted it's not an "object" in that sense of the word, it's a "mathematical object". And I insisted that it ought not be called an object of any sort. So you proceeded with an unacceptable interpretation of the law of identity in an attempt to validate your claim. What I believe, is that "mathematical object" is an incoherent concept.

What's incoherent is you objecting to 4 = 4 as an instance of the law of identity; and claiming that 2 + 2 and 4 refer to different mathematical objects, when in fact I showed you a formal proof that they represent exactly the same mathematical object. A proof that you refuse to this day to even acknowledge, let alone refute or discuss. That's to your shame. You pretend to be an honest conversant but you are not.

This depends on what = represents.

Bill Clinton on his slimiest day never reached such heights of bullshit. Pardon my French.

Does it represent "is the same as", or does it represent "is equal to"?

There's no difference. And you have failed to articulate even a plausible argument for a difference.

From our last discussion, you did not seem to respect a difference between the meaning of these two phrases.

You got that right.

If you're still of the same mind, then there is no point in proceeding until we work out this little problem.

A point on which we agree. There's no point in proceeding. You're trolling me as you have been for two years now.

This is why I say context of the symbol is important. When the law of identity is represented as a=a, = symbolizes "is the same as". But when we write 2+2=4, = symbolizes "is equal to". If we assume that a symbol always represents the very same thing in every instance of usage, we are sure to equivocate. Clearly, "is the same as" does not mean the same thing as "is equal to".

Bullpucky to da max.

No, of course not, that's clearly a false representation of what math is. That would be like saying that 2+2=4 could be considered to be valid regardless of what the symbols mean. That's nonsense, it's what the symbols mean which gives validity to math.

So you're right and Hilbert and Euclid are wrong. Ok. Forgive me if I choose to disagree.

Yes, that is exactly the case it can even mean something different in the same sentence. When some says "I reserved a table for 4 at 4", each instance of 4 means something different to me.

LOL. That's a good example and I take your point. Natural language is often ambiguous. In a given mathematical context, a given symbol holds the exact same meaning throughout. Dinner isn't a mathematical context. But yeah that's a good example of the problem with ambiguity of natural language. And thanks for the chuckle.

And, as I explained to you already, when someone says 2+2=4, each instance of 2 must refer to something different or else there would not be four, only two distinct instances of the very same two, and this would not make four.

That's utter nonsense. Interestingly though, I came across this identical line of reasoning this morning. I don't know what the source or full context of this page is, but you either have a kindred spirit, or you wrote this. Author is objecting to the equation 2 = 1 + 1. He says:

The existence of two requires two copes of one. Then there are two objects which are separable and yet absolutely indistinguishable.

But I insist that this appeal to eternal abstract beings which exist in copies is illicit. If you claim the existence of two copies of one, what differentiates them? There would have to be a feature to distinguish them; yet arithmetic insists there is no such feature. As eternal abstract beings they have to be identical in every respect. Then how can they be plural? I conclude that there can’t be identical copies of the pure number one.

Hard to believe there are two people who assert this nonsense, not just you alone. Unless as I say you are the author. But I concede that IF you are not the author, then you are not unique in your confusion regarding mathematical objects.

http://www.henryflynt.org/meta_tech/that1=2.html

By the way there is a standard formalism for obtaining multiple copies of the same object, you just Cartesian-product them with a distinct integer. So if you need two copies of the real line $\mathbb R$, you just take them as $\mathbb R \times \{1\}$ and $\mathbb R \times \{2\}$. It's not that mathematicians haven't thought about this problem. It's that they have, and they have easily handled it. As usual you confuse mathematical ignorance with philosophical insight.

In any event, you avoided (as you always do when presented with a point you can't defend) my question. If set theorists are not only wrong but morally bad, is Euclid equally so? You stand by your claim that set theorists are morally bad? Those are your words. Defend or retract please.

Clearly, "is the same as" does not mean the same thing as "is equal to".
— Metaphysician Undercover

I say it does. I think you're splitting hairs for the sake of argument.

Thanks man, sometimes @Meta makes me question my own sanity.
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My interpretation of the question of universal and particular is close to Thomism, as I currently understand it. Going back to Greek dialectic, the origin of the idea is the debate about the reliability of the testimony of sense, and the related question about the nature of knowledge (episteme). The salient point in this context is that sensory perception and its objects do not possess the same degree of apodictic certainty as do arithmetical and geometric judgements. The latter are known directly by 'the eye of reason' and are not mediated by the sensory organs. Same with the forms of things - these are like the ideas or principles that give rise to particulars, and they are apprehended by 'nous' rather than by the sense. So knowledge and judgement comprise the synthesis of universals, which are the objects of pure intuition (or nous, hence, noumenal) and particulars, which is the physical form apprehended by the physical senses. Particulars are real insofar as they're instantiations of the idea, which is their unchanging form; that is the sense in which the ideas 'lend being' to particulars, or particulars are said to 'participate in' the form.

In Plato's dialogues, whenever metaphysical ideas are encountered, they are nearly always referred to as 'likely stories'and treated with a degree of diffidence. Plato was never close to a naive or even scientific realist, many of the ideas in Platonic dialogues are obviously mythological, symbolic, or allegorical. Accordingly, I don't claim to know whether the Platonic forms are real. But I will say, that this 'traditionalist' account, preserved mainly in various forms of Christian Platonism, is a plausible and coherent philosophical view which makes much more sense than modern irrationalism. (Of course, from the modern p-o-v, this is simply clinging to a nostalgic and bygone philosophy, but I'm ok with that.)
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I say it does. I think you're splitting hairs for the sake of argument.

Are you serious? As human beings, you and I are equal, based in a principle of equality. Clearly we are not the same. A judgement of equality is based in a principle of measurement, volume, weight, temperature, species, whatever. It allows that two distinct things are equal, by the precepts of the principle. They are the same volume, or the same weight, the same temperature, or the same species. Notice how in the concept of "equal", "the same" is qualified so that it is what is attributed to the thing, volume, weight, etc., which is said to be the same, not the thing itself. Under the law of identity, a thing is the same as itself. So it is impossible that two distinct things are the same thing, as we say that two distinct things are equal. By that law, we can only use "the same" to refer to one and the same thing, the very same thing. This is not a matter of splitting hairs, there's a fundamental difference between two distinct things which are the same in some way (equal by that principle), and one thing, of which no other thing can be said to be the same thing as.

My mind has blocked them out as traumatic experiences.

On the basis of that statement I am concluding that proceeding with any discussion with you on this matter is pointless because your mind is liable to block out anything I write.

What's incoherent is you objecting to 4 = 4 as an instance of the law of identity

You obviously do not know the law of identity. I had to spell it out for you already. You objected, and offered some axiom of equality which is obviously not the law of identity. This statement above, indicates that you clearly did not take the time to learn it yet. From what I learned last time, until we agree as to what the law of identity stipulates, further discussion on this issue is pointless.

In a given mathematical context, a given symbol holds the exact same meaning throughout.

So the issue is, in the context of mathematics, does = mean equal to, or does it mean the same as? I'm sure you can grasp the fact that you and I are equal, as human beings, but we are not the same as each other. Therefore, I'm sure you can accept that equal to has a different meaning from the same as. Which does = symbolize in the context of mathematics?

Hard to believe there are two people who assert this nonsense, not just you alone.

Reason is contagious, it tends to catch on. Notice that the op agrees with me as well. And, I think jgill agreed with me on this point in that other thread as well. I don't understand why the obvious appears as nonsense to you. It's very clear, that if 1 always referred to the same object we could not make 2 out of two instances of 1, we'd always have just one object symbolized.

By the way there is a standard formalism for obtaining multiple copies of the same object, you just Cartesian-product them with a distinct integer. So if you need two copies of the real line RR, you just take them as R×{1}R×{1} and R×{2}R×{2}. It's not that mathematicians haven't thought about this problem. It's that they have, and they have easily handled it. As usual you confuse mathematical ignorance with philosophical insight.

It doesn't matter how you formalize it, the point is that it violates the law of identity. "Multiple copies of the same object" is exactly what is outlawed. You can rationalize your violation of any law however you like, but it doesn't change the fact that you violate the law. You can show me a thousand objects, and insist that according to your axioms they are all one and the same object. So what? All this indicates is that your axioms are faulty.

In any event, you avoided (as you always do when presented with a point you can't defend) my question. If set theorists are not only wrong but morally bad, is Euclid equally so? You stand by your claim that set theorists are morally bad? Those are your words. Defend or retract please.

Did I say set theorists are morally bad? No, it was an analogy. The analogy was that if I saw set theorists doing something I thought was wrong (bad), I might be inclined toward explaining to them how I thought what they are doing is wrong, just like if I saw someone behaving in a greedy way which I thought was morally wrong, I might be inclined to explain to them why I thought what they were doing is morally wrong. The point being that it really doesn't make very much difference to me, in my life personally, if these people, either the set theorists, or the greedy immoral people, continue along their misguided pathways. Nevertheless, I might take it upon myself to make an attempt to point out to them how their pathways are misguided.

Particulars are real insofar as they're instantiations of the idea, which is their unchanging form; that is the sense in which the ideas 'lend being' to particulars, or particulars are said to 'participate in' the form.

This is where Aristotle parts from Plato. In Plato's Timaeus particulars are supposed to be in some way derived from universal forms. But Plato is incapable of describing the mechanism by which a universal form could create the existence of a particular individual. He found the need to posit "matter" as the recipient of the form, in order to account for the particularities of the individual. The peculiarities of the individual are due to the matter. But when Aristotle developed this idea he discovered that matter itself cannot account for any of the properties of an object, and so each individual thing must have a unique form proper to itself. That's his hylomorphism

This was the fundamental question of his metaphysics, why is a thing the unique thing which it is, rather than something else. He said the commonly asked question of why there is something rather than nothing cannot be answered, and is therefore a fruitless question. So he suggested the proper question to ask of being qua being, is why is there what there is instead of something else. Why is each thing the unique and particular thing that it is, instead of something else. This led him to the conclusion that there is a unique and particular form which is responsible for each thing being the particular thing which it is. Hence the law of identity as formulated, each thing has a unique identity, it is the same as itself, and nothing else. For Aristotle, this is the reality of the particular.
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